We want models with uniform finite approximate closure because, as the bounding constant goes to zero, \(B \rightarrow 0\) , we may use \(K\)  to perform stability, observability and spectral analysis. We see this is true as, in the discrete-time formulation, as \(B\rightarrow 0\) , \(K\)  approaches a projection of the action of the Koopman operator [1]}. When our dictionary is state-inclusive, it is trivial to project from from \(\psi \)  to \(y\)  and its trajectory may yield stability insights. To keep our models meaningful in this way, all the dictionaries in this article are state-inclusive.
